Stat 567:Statistical Reliability
Ninth Class (October 1, 1997)

Maximum Likelihood Estimation

Class Objectives:

• Learn to calculate standard errors of Maximum Likelihood Estimators (MLEs) and their covariance matrix

Homework Assignments:

• Redo the questions you missed in the midterm exam. Be neat! Append you solutions to the exam and turn this package in. You will receive a homework grade for this work.

Class Outline and Main Points:

• Asymptotic theory of maximum likelihood estimation
  • For "large samples" and mild regularity conditions MLEs are approximately normal, approximately unbiased, and have the smallest attainable variance among estimators
  • Information matrix
  • Variance-Covariance matrix
• Exponential distribution with right censoring
  • Estimators for the hazard rate and mean time to failure
  • Approximate standard error of estimator

Using JMP to obtain MLEs:

To get the MLEs with JMP of the Weibull and Gumbel (Extreme-value) fits of the ball bearing data of Example 2.1, Page 37 of your textbook go to JMP's survival platform and obtain the P-L survival estimate as we have done before. Then proceed as follows:

To get MLEs for a log-normal model of the ball bearing data proceed as above but select the log-normal plot instead of the Weibull plot.

Notice that with this JMP platform we do not get the variance-covariance matrix of the estimates. Later we will show how to get this matrix using JMP's non-linear platform.

Study Questions:

• Of what use is the variance-covariance matrix?
• What are some of the properties of ML estimators that make them desirable
• Why do we need numerical techniques to obtain MLEs?

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